I am sorry to hear you are worried about Precious. My name is Rebecca. I am a veterinarian and will do my best to help.
Garlic and other members of the onion family can be toxic to dogs. Small doses are safe, but in general I do not recommend giving onions or garlic to dogs. Since she is a small breed, it would be easy to give her too much.
Her cough could be from many things: an infection, bad teeth, allergies, heart disease or lung disease. It is always best to find the underlying cause, so you know the proper treatment. How do her teeth look?
How much does she weigh? It would be safe to try Benadryl for allergies (plain Benadryl, not Benadryl D), or Robitussin DM (only the DM) for coughing. However, if she were my patient or my own dog, I would take her temperature, and listen to her heart and lungs and look at her teeth before deciding the best treatment.
Let me know what other questions you have and if this is helpful please.